Pre-Algebra Examples

$\frac{x}{x - 3} - \frac{x - 4}{x + 3}$

Step 1

To write $\frac{x}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{x}{x - 3} \cdot \frac{x + 3}{x + 3} - \frac{x - 4}{x + 3}$

Step 2

To write $- \frac{x - 4}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x}{x - 3} \cdot \frac{x + 3}{x + 3} - \frac{x - 4}{x + 3} \cdot \frac{x - 3}{x - 3}$

Step 3

Write each expression with a common denominator of $(x - 3) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{x}{x - 3}$ by $\frac{x + 3}{x + 3}$ .

$\frac{x (x + 3)}{(x - 3) (x + 3)} - \frac{x - 4}{x + 3} \cdot \frac{x - 3}{x - 3}$

Step 3.2

Multiply $\frac{x - 4}{x + 3}$ by $\frac{x - 3}{x - 3}$ .

$\frac{x (x + 3)}{(x - 3) (x + 3)} - \frac{(x - 4) (x - 3)}{(x + 3) (x - 3)}$

Step 3.3

Reorder the factors of $(x - 3) (x + 3)$ .

$\frac{x (x + 3)}{(x + 3) (x - 3)} - \frac{(x - 4) (x - 3)}{(x + 3) (x - 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{x (x + 3) - (x - 4) (x - 3)}{(x + 3) (x - 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 3 - (x - 4) (x - 3)}{(x + 3) (x - 3)}$

Step 5.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 3 - (x - 4) (x - 3)}{(x + 3) (x - 3)}$

Step 5.3

Move $3$ to the left of $x$ .

$\frac{x^{2} + 3 \cdot x - (x - 4) (x - 3)}{(x + 3) (x - 3)}$

Step 5.4

Apply the distributive property.

$\frac{x^{2} + 3 x + (- x - - 4) (x - 3)}{(x + 3) (x - 3)}$

Step 5.5

Multiply $- 1$ by $- 4$ .

$\frac{x^{2} + 3 x + (- x + 4) (x - 3)}{(x + 3) (x - 3)}$

Step 5.6

Expand $(- x + 4) (x - 3)$ using the FOIL Method.

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Step 5.6.1

Apply the distributive property.

$\frac{x^{2} + 3 x - x (x - 3) + 4 (x - 3)}{(x + 3) (x - 3)}$

Step 5.6.2

Apply the distributive property.

$\frac{x^{2} + 3 x - x \cdot x - x \cdot - 3 + 4 (x - 3)}{(x + 3) (x - 3)}$

Step 5.6.3

Apply the distributive property.

$\frac{x^{2} + 3 x - x \cdot x - x \cdot - 3 + 4 x + 4 \cdot - 3}{(x + 3) (x - 3)}$

Step 5.7

Simplify and combine like terms.

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Step 5.7.1

Simplify each term.

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Step 5.7.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.7.1.1.1

Move $x$ .

$\frac{x^{2} + 3 x - (x \cdot x) - x \cdot - 3 + 4 x + 4 \cdot - 3}{(x + 3) (x - 3)}$

Step 5.7.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} + 3 x - x^{2} - x \cdot - 3 + 4 x + 4 \cdot - 3}{(x + 3) (x - 3)}$

Step 5.7.1.2

Multiply $- 3$ by $- 1$ .

$\frac{x^{2} + 3 x - x^{2} + 3 x + 4 x + 4 \cdot - 3}{(x + 3) (x - 3)}$

Step 5.7.1.3

Multiply $4$ by $- 3$ .

$\frac{x^{2} + 3 x - x^{2} + 3 x + 4 x - 12}{(x + 3) (x - 3)}$

Step 5.7.2

Add $3 x$ and $4 x$ .

$\frac{x^{2} + 3 x - x^{2} + 7 x - 12}{(x + 3) (x - 3)}$

Step 5.8

Subtract $x^{2}$ from $x^{2}$ .

$\frac{3 x + 0 + 7 x - 12}{(x + 3) (x - 3)}$

Step 5.9

Add $3 x$ and $0$ .

$\frac{3 x + 7 x - 12}{(x + 3) (x - 3)}$

Step 5.10

Add $3 x$ and $7 x$ .

$\frac{10 x - 12}{(x + 3) (x - 3)}$

Step 5.11

Factor $2$ out of $10 x - 12$ .

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Step 5.11.1

Factor $2$ out of $10 x$ .

$\frac{2 (5 x) - 12}{(x + 3) (x - 3)}$

Step 5.11.2

Factor $2$ out of $- 12$ .

$\frac{2 (5 x) + 2 (- 6)}{(x + 3) (x - 3)}$

Step 5.11.3

Factor $2$ out of $2 (5 x) + 2 (- 6)$ .

$\frac{2 (5 x - 6)}{(x + 3) (x - 3)}$